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Multiobjective with single objectives that use distinct components of input vec? What does it mean for other components in input vec?

E.g.

$$\min (x_1,x_2)$$ $$x \in \mathbb{R}^2, x_1,x_2\geq0$$

Then if one solves this by one objective at a time, then one solves first $x_1$ then $x_2$. However, the input vector to this problem is $x \in \mathbb{R}^2$.

One could e.g. use initial guess $(0.5,0.5)$. Then the algorithm (e.g. minimize in scipy) might spit out $(0.0, 0.43)$. Since we were minimizing the first objective $x_1$, then the solution that we're interested in is $0.0$. However, the algorithm has clearly varied the second component as well, ending to $0.43$ due to some things in the algo.

Is the $0.43$ redundant? Why is it altered by e.g. scipy's minimize, even when the 1st objective doesn't contain it.

mavavilj
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  • Can you clarify your question? It sounds like you are asking how to solve an optimization problem for a vector $(x_1, x_2, ..., x_n)$ when only a subset of those variables appear in the objective. Yet, your example does not do that, it has $(x_1,x_2)$ appearing in both the objective and the constraints. Anyway, it is common for the objective to only use a subset of the variables. You still need to solve for the entire vector of all variables because the variables in the objective may be coupled with the other variables through the constraints. – Michael Apr 06 '19 at 18:47
  • I also do not know what you mean by "solving first $x_1$ but inputting $x=(x_1,x_2)$." It is also not clear if you are trying to minimize or maximize your objective function $\min[x_1,x_2]$. I assume you are trying to minimize $\min[x_1,x_2]$ subject to the constraints, in which case the solutions are $(0,a)$ or $(a,0)$ with $a\geq 0$. – Michael Apr 06 '19 at 18:50

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